Statistics – 3

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59 Statistics – 3

Chittaranjan Andrade

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1. ANOVA is appropriate in a study with a repeated measures design. image image
2. In a double-blind controlled study on the effect a new drug has on anxiety levels, patients are not told whether they are receiving drug or placebo. image image
3. The chi-squared test could be used in a study of comparison of pay vs. job satisfaction in two groups of workers, one male and the other female. image image
4. Chi-squared could be used to analyse the results of a questionnaire measuring level of anxiety in a random sample of the population. image image
5. The variation in a variable can be explained by the square of its correlation coefficient with another variable. image image
6. The closer the correlation coefficient ‘r’ is to zero, the fewer errors made in prediction. image image
7. In an IQ test showing normal distribution with a mean of 100 and a standard deviation of 15, approximately 15% of the population have an IQ less than 70. image image
8. The Fisher exact probability test should not be used for small numbers. image image
9. Incidence includes cases that are no longer current. image image
10. Non-parametric statistics can be measured by least squares. image image
11. The Mann–Whitney test could be used to analyse the results of a questionnaire measuring level of anxiety in a random sample of the population. image image
12. The mode is a measure of central tendency. image image
13. In an experiment to determine whether drug therapy or hypnosis is better in treatment to stop smoking, the results are measured by the number of cigarettes smoked after treatment. In this study the null hypothesis postulates there will be a significant difference between drug therapy and hypnosis in reduction of cigarette smoking. image image
14. A population pyramid can be used to describe the sex and age structure of a population. image image
15. The primacy effect includes deciding about a person by their physical appearance. image image
16. In categorical independent variables, regression is usually by the method of least squares. image image
17. Scores obtained on two successive administrations correlated 0.90 indicate reliability and not validity of a personality questionnaire. image image
18. A new ‘treatment’ is tried on gifted children, all of whom have an IQ greater than 140. If the mean IQ of the sample is 150, and if the treatment has no effect on IQ, the mean IQ that one would expect on re-testing 1 year later is slightly greater than 150. image image
19. The telephone directory can be used for random sampling. image image
20. Standard deviation is a measure of precision of an estimate. image image
21. A statement that an association between two variables is statistically significant means that, if the study is repeated, chances are that the significance of the association would be the same. image image
22. A result statistically significant at a level of 5% implies that there is a 5% chance that the results suggest a wrong conclusion. image image
23. Stratification increases the power of a study. image image
24. The T2 test is used for non-parametric variables. image image
25. In an experiment designed to investigate the effect of home background on intelligence, the family size is a dependent variable. image image

ANSWERS

1. ANOVA is appropriate in a study with a repeated measures design.

True: A repeated measures design is one in which subjects are assessed at two or more time points. If there is only one group and only two time points, and if the dependent variable is a continuous variable, the paired t-test is the simplest procedure. However, a model of ANOVA known as repeated measures ANOVA can also be performed. It is the only test possible when there is more than one group or more than two time points.

If there are only two assessment points, the mixed-model procedure is used. If there are more than two assessment points, the assumptions of the mixed-model (sphericity, homogeneity of variances, etc.) are often violated, and repeated measures multivariate ANOVA is the more appropriate procedure. It can be executed using the Hotelling’s T-square, Pillai’s, Wilk’s, or Roy’s procedures.

2. In a double-blind controlled study on the effect a new drug has on anxiety levels, patients are not told whether they are receiving drug or placebo.

True: Single-blind studies are those in which the investigator does not know the experimental condition to which the subject belongs, but the subject is aware of the assignment. Double-blind studies are those in which neither subject nor investigator is aware of the treatment assignment. Subject blindness is necessary to ensure that subject expectations about the treatment do not bias their response, or their report of response. Investigator blindness is necessary to ensure that investigator expectations from treatment or pressures on the investigator do not contaminate their assessments.

3. The chi-squared test could be used in a study of comparison of pay vs. job satisfaction in two groups of workers, one male and the other female.

False: Pay and job satisfaction are usually both measured as continuous (quantitative) variables along a ratio scale. The relationship between pay and job satisfaction would be tested using a method of correlation. However, if we wished to determine whether the relationship between pay and job satisfaction differs between males and females, we would need to use logistic regression with gender as the dependent (grouping) variable, and with pay and job satisfaction as the independent variables. We would need to specify an interaction term between pay and job satisfaction.

4. Chi-squared could be used to analyse the results of a questionnaire measuring level of anxiety in a random sample of the population.

False: There is no hypothesis stated here. All that can be done is to compute descriptive statistics, such as range, mean and standard deviation, median, mode, etc.

5. The variation in a variable can be explained by the square of its correlation coefficient with another variable.

True: If the correlation between age and speed of performance on a test is 0.6, the variance is 0.36, i.e. 36%. This means that 36% of the variance (variation) in speed of performance is explained by variations in age. This variance should not be confused with variance which is the square of the standard deviation, which is used in computing statistics such as the t-value or the F value.

6. The closer the correlation coefficient ‘r’ is to zero, the fewer errors made in prediction.

False: The closer the correlation coefficient is to zero, the weaker is the relationship between the variables, and hence the less accurately does the independent variable predict the dependent variable.

7. In an IQ test showing normal distribution with a mean of 100 and a standard deviation of 15, approximately 15% of the population have an IQ less than 70.

False: In a normal distribution, 68.3% of the population lie within 1 standard deviation of the mean, i.e. between 85 and 115; 95.5% lie within 2 standard deviations of the mean, i.e. between 70 and 130. Therefore, 4.5% lie below 70 and above 130, or, 2.25% lie below 70 and 2.25% lie above 130. Moreover, if 68.3% lie between 85 and 115, and if 95.5% lie between 70 and 130, 27.2% (95.5–68.3) lie between 70 and 85, and between 115 and 130. Or, 13.6% lie between 70 and 85, and 13.6% lie between 115 and 130.

In this example, if the mean is 100, the median and mode are also 100 because the distribution is normal.

8. The Fisher exact probability test should not be used for small numbers.

False: The chi-squared test has certain assumptions. One of these is that the expected frequency in at least 80% of cells should be 5 or above, and no expected frequency should be <1. In this context, expected frequency for a cell is given by the product of the row and column totals (for that cell) divided by the grand total (that is, the sample size).

In 2 × 2 and 2 × 3 chi-squared contingency tables, if the expected frequency assumption is not met, the Fisher’s exact test can be used. Some researchers routinely use the Fisher’s exact test if the sample size is <20.

The Fisher’s exact test is far more computationally intensive than the chi-squared test.

9. Incidence includes cases that are no longer current.

False: Incidence refers to the occurrence of new cases in a population during a specified period of time. Incidence is usually described as the number of new cases per 100 (or 1000, or 100 000, as appropriate) population.

10. Non-parametric statistics can be measured by least squares.

True: Here, Spearman’s procedure is the non-parametric equivalent of Pearson’s procedure. Spearman’s procedure is conducted on data that have been ranked rather than on the raw data. Spearman’s procedure is applied when the distribution of the raw data is non-normal, or when the primary data were collected as ranked data.

11. The Mann–Whitney test could be used to analyse the results of a questionnaire measuring level of anxiety in a random sample of the population.

False: The Mann–Whitney test is a non-parametric procedure which compares a continuous variable between two groups when the assumptions for the t-test are not met.

In the question above, no groups are specified. Had two groups been specified (e.g. males vs. females), and had the distributions of anxiety scores been non-normal, the Mann–Whitney test could have been used.

12. The mode is a measure of central tendency.

True: The mean, median and mode are measures of central tendency.

There are different kinds of mean, i.e. arithmetic, geometric, and harmonic. In statistics, we almost always consider the arithmetic mean, or average, which is the sum of the terms divided by the number of the terms.

The median is the middle term when the terms are arranged in ascending or descending order of magnitude. If the number of terms in the sample is even, the median is the average of the middle two terms.

The mode is the most commonly occurring term in the sample.

The mean is used for most statistical descriptions and procedures. Tests of the mean are more powerful, i.e. less vulnerable to a type II error.

The median is used when the distribution is non-normal. The mode is used when we are most interested in the frequently occurring values.

13. In an experiment to determine whether drug therapy or hypnosis is better in treatment to stop smoking, the results are measured by the number of cigarettes smoked after treatment. In this study the null hypothesis postulates there will be a significant difference between drug therapy and hypnosis in reduction of cigarette smoking.

False: The null hypothesis states that there will be no difference in the reduction in cigarette smoking between the two treatments. It is the alternative hypothesis which states that the two treatments will differ significantly in their effect on smoking.

14. A population pyramid can be used to describe the sex and age structure of a population.

True: The X-axis describes the population and the Y-axis describes age groups. A separate graph is formed for males and for females, and a mirror image of one is placed adjacent to the original graph of the other to produce the shape of a pyramid.

15. The primacy effect includes deciding about a person by their physical appearance.

False: The primacy effect is when an initial situation influences the interpretation or effect of a later one. For example, we often make persistent judgements about people based upon first impressions. This is a classic example of the primacy effect.

In contrast, the recency effect is when a recent situation is associated with a greater influence than an earlier situation. For example, if we are asked to remember a word list, we most easily remember the words which were read out last.

16. In categorical independent variables, regression is usually by the method of least squares.

True: In a multiple regression procedure a dichotomous categorical independent variable will be coded as 0 or 1. If the variable is divided into more than two categories, dummy variables are created, and the presence or absence of each of the categories is coded as 0 or 1. Incidentally, ANOVA can be thought of as linear regression with coded categorical independent variables. Like simple linear regression, multivariate regression makes use of the least squares technique.

17. Scores obtained on two successive administrations correlated 0.90 indicate reliability and not validity of a personality questionnaire.

True: Reliability means replicability. A reliable instrument is one which yields the same or very similar results no matter how many times it is administered to the same individual (test–retest reliability) or no matter how many different persons administer it to the same individual (inter-rater reliability).

There are different methods to assess reliability, and correlation is not one of the recommended ones!

18. A new ‘treatment’ is tried on gifted children, all of whom have an IQ greater than 140. If the mean IQ of the sample is 150, and if the treatment has no effect on IQ, the mean IQ that one would expect on re-testing 1 year later is slightly greater than 150.

False: If the treatment has no effect, the mean IQ should remain the same, i.e. 150. However, as it is unrealistic to expect that the children will perform in exactly the same manner as they did the previous year, the mean IQ at re-testing could be either above 150 or below 150 but not significantly different from 150. Note that as the variation in performance on re-testing is random, we cannot expect that the mean on re-testing will be greater than the baseline mean.

19. The telephone directory can be used for random sampling.

False: Persons who own a telephone and who are listed in the directory comprise a socio-economic subset of the population and do not represent the population. Therefore, a sample drawn from the telephone directory will be a biased sample.

The telephone directory can be used for random sampling only if the population is defined as all those who own telephones. This, for example, is meaningful if the subject of the study is telephone behaviour or usage.

If the telephone directory is used for recruitment, it must be recognized that this is a convenience sample. In a way, almost all studies are conducted on convenience samples. For example, every sample drawn in a hospital setting is a convenience sample because patients who come to the hospital are likely to systematically differ from those in the general population. The only way to obtain a truly random sample is to use population-based methods of sampling, as in epidemiological studies.

20. Standard deviation is a measure of precision of an estimate.

False: The standard deviation is a measure of dispersion. That is, it is a measure of the extent of scatter of the individual terms (in the sample) around the mean. The mean of a sample is only an estimate of the population mean. The standard error of the mean, and the confidence intervals for the mean, are measures of the accuracy of the mean.

21. A statement that an association between two variables is statistically significant means that, if the study is repeated, chances are that the significance of the association would be the same.

True: If two variables are significantly associated and we repeat our study, we should find a significant association 95 times out of 100.

22. A result statistically significant at a level of 5% implies that there is a 5% chance that the results suggest a wrong conclusion.

True: It means that there is a 5% probability that the result is due to chance factors, and a 95% probability that the result is due to genuine reasons.

23. Stratification increases the power of a study.

True: This is because stratification seeks to balance the groups on important, potentially confounding (independent) variables. As a result, variability resulting from heterogeneity across groups is diminished. As power is inversely related to variability, reducing the variability through stratification will increase the power of the study.

24. The T2 test is used for non-parametric variables.

False: The Hotelling’s T2 test is one of the ways of executing a multivariate analysis of variance (MANOVA). In this test, several means are simultaneously compared between groups to reduce the risk of a false positive (type 1) error which may arise if each mean is compared separately between the groups.

25. In an experiment designed to investigate the effect of home background on intelligence, the family size is a dependent variable.

False: Family size is an independent variable and intelligence is the dependent variable. Other home background characteristics which may be examined as independent variables include parental intelligence, parental aspirations for the child, parental socio-economic status, etc. Dependent variables could be the child’s IQ score, the child’s examination results, etc.

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